Direct numerical simulations of turbulent non-rotating and rotating plane Couette flow with a periodically modulated plate velocity are conducted to study the effect of modulated forcing on turbulent shear flows. The time-averaged shear Reynolds number is fixed at $Re_S = 3 \times 10^4$, which results in a frictional Reynolds number of approximately $Re_\tau \approx 400$. The modulating frequency is varied in the range $Wo\in (20,200)$, while the modulating amplitude is kept fixed at $10\,\%$ of the shear velocity except to demonstrate that varying this parameter has minimal effect. The resulting shear at the plates is found to be independent of the forcing frequency, and equal to the non-modulated baseline. For the non-rotating simulations, two clear flow regions can be seen: a near-wall region that follows Stokes’ theoretical solution, and a bulk region that behaves similar to Stokes’ solutions but with an increased effective viscosity. For high driving frequencies, the amplitude response follows the scaling laws for modulated turbulence of von der Heydt et al. (Phys. Rev. E, vol. 67, 2003, 046308). Cyclonic rotation is not found to modify the system's behaviour in a substantial way, but anti-cyclonic rotation changes significantly the system's response to periodic forcing. We find that the persistent axial inhomogeneities introduced by mild anti-cyclonic rotation make it impossible to measure the propagation of the modulation adequately, while stronger anti-cyclonic rotation creates regions where the modulation travels instantaneously.

The rules that govern a two-dimensional inviscid incompressible fluid are simple. Yet, to characterise the long-time behaviour is a knotty problem. The fluid fulfils Euler's equations: a nonlinear Hamiltonian system with an infinite number of conservation laws. In both experiments and numerical simulations, coherent vortex structures emerge after an initial stage. These formations dominate the large-scale dynamics, but small scales also emerge and persist. The resulting scale separation resembles Kraichnan's theory of forward and backward cascades of enstrophy and energy. Previous attempts to model the double cascade use filtering techniques that enforce separation from the outset. Here, we show that Euler's equations possess an intrinsic, canonical splitting of the vorticity function. The splitting is remarkable in four ways: (i) it is defined solely by the Poisson bracket and the Hamiltonian; (ii) it characterises steady flows; (iii) it innately separates scales, enabling the dynamics behind Kraichnan's qualitative description; and (iv) it accounts for ‘broken line’ energy spectra observed in both experiments and numerical simulations. The splitting originates from Zeitlin's truncated model of Euler's equations in combination with a standard quantum tool: the spectral decomposition of Hermitian matrices. In addition to theoretical insight, the scale separation dynamics enables stochastic model reduction, where multiplicative noise models small scales.

Partially wetting droplets under an airflow can exhibit complex behaviours that arise from the coupling of surface tension, inertia of the external flow and contact-line dynamics. Recent experiments by Hooshanginejad et al. (J. Fluid Mech., vol. 901, 2020) revealed that a millimetric partially wetting water droplet under an impinging jet can oscillate in place, split or depin away from the jet, depending on the magnitude (i.e. $5\unicode{x2013}20\ {\rm m}\ {\rm s}^{-1}$) and position of the jet. To rationalise the experimental observations, we develop a two-dimensional lubrication model of the droplet that incorporates the external pressure of the impinging high-Reynolds-number jet, in addition to the capillary and hydrostatic pressures of the droplet. Distinct from the previous model by Hooshanginejad et al. (J. Fluid Mech., vol. 901, 2020), we simulate the motion of the contact line using precursor film and disjoining pressure, which allows us to capture a wider range of droplet behaviours, including the droplet dislodging to one side. Our simulations exhibit a comparable time-scale of droplet deformations and similar outcomes as the experimental observations. We also obtain the analytical steady-state solutions of the droplet shapes and construct the minimum criteria for splitting and depinning.

This paper presents bifurcation analyses characterising the nonlinear dynamics of fully developed laminar annular jets with respect to the centrebody diameter ${ {d}}$, Reynolds number ${ {Re}}$, and swirl ratio ${ {S}}$. Similar flows appear in numerous applications and feature a vibrant range of topological and dynamical characteristics associated with phenomena including shear layer separation and vortex breakdown. Our results begin by describing the non-monotonic evolution of the axisymmetric jet's steady topology under varying ${ {S}}$. In accord with earlier reports, the jet progresses through a sequence of wake, breakdown and wall jet regimes in a qualitatively similar manner across a wide span of ${ {d}}$ and ${ {Re}}$ values. In the wake regime, the non-swirling jet bifurcates to a plane-symmetric, but not axisymmetric, steady flow pattern beyond a ${ {d}}$-dependent critical ${ {Re}}$ value. With further increase in ${ {Re}}$, the steady non-swirling jet destabilises subsequently via multiple distinct Hopf bifurcations. Introducing ${ {S}}>0$ to the jet also induces unsteadiness by twisting the singly azimuthally periodic ($|m|=1$) asymmetric wake structure and causing it to precess periodically in time about the central axis. Intermediate swirl stabilises this unsteady dynamics and restores the jet's axisymmetry. This stabilising effect is then reversed in the breakdown regime at higher ${ {S}}$, where a variety of different $|m|=1$ and $|m|=2$ instabilities bifurcate from the steady flow as ${ {S}}$ is increased. Several instances of hysteresis and subcritical behaviour are reported and discussed, including one that manifests precessing vortex core oscillations.

Container motion along a planar circular trajectory at a constant angular velocity, i.e. orbital shaking, is of interest in several industrial applications, e.g. for fermentation processes or in cultivation of stem cells, where good mixing and efficient gas exchange are the main targets. Under these external forcing conditions, the free surface typically exhibits a primary steady-state motion through a single-crest dynamics, whose wave amplitude, as a function of the external forcing parameters, shows a Duffing-like behaviour. However, previous experiments in laboratory-scale cylindrical containers have revealed that, owing to the excitation of super-harmonics, diverse dynamics are observable in certain driving-frequency ranges. Among these super-harmonics, the double-crest dynamics is particularly relevant, as it displays a notably large amplitude response, which is strongly favoured by the spatial structure of the external forcing. In the inviscid limit and with regards to circular cylindrical containers, we formalize here a weakly nonlinear analysis via a multiple-time-scale method of the full hydrodynamic sloshing system, leading to an amplitude equation suitable for describing such a double-crest swirling motion. The weakly nonlinear prediction is shown to be in fairly good agreement with previous experiments described in the literature. Lastly, we discuss how an analogous amplitude equation can be derived by solving asymptotically for the first super-harmonic of the forced Helmholtz–Duffing equation with small nonlinearities.

A two-dimensional model for the evolution of the fire line – the interface between burned and unburned regions of a wildfire – is formulated. The fire line normal velocity has three contributions: (i) a constant rate of spread representing convection and radiation effects; (ii) a curvature term that smooths the fire line; and (iii) a Stefan-like term in the direction of the oxygen gradient. While the first two effects are geometrical, (iii) is dynamical and requires the solution of the steady advection–diffusion equation for oxygen, with advection owing to a self-induced ‘fire wind’, modelled by the gradient of a harmonic potential field. The conformal invariance of this coupled pair of partial differential equations, which has the Péclet number $\textit {Pe}$ as its only parameter, is exploited to compute numerically the evolution of both radial and infinitely long periodic fire lines. A linear stability analysis shows that fire line instability is possible, dependent on the ratio of curvature to oxygen effects. Unstable fire lines develop finger-like protrusions into the unburned region; the geometry of these fingers is varied and depends on the relative magnitudes of (i)–(iii). It is argued that for radial fires, the fire wind strength scales with the fire's effective radius, meaning that $\textit {Pe}$ increases in time, so all fire lines eventually become unstable. For periodic fire lines, $\textit {Pe}$ remains constant, so fire line stability is possible. The results of this study provide a possible explanation for the formation of fire fingers observed in wildfires.

We study experimentally the behaviour of negatively buoyant disks and fibres in a turbulent boundary layer. The regime is relevant to the transport of natural sediment or plastic particles in water, with density ratio $\rho _p/\rho _f \sim O(1)$, major axis lengths $D_p^+ \sim 50$, friction Stokes numbers $\mbox { {St}}^+ \sim O(10)$ and friction Reynolds number $\mbox { {Re}}_\tau = 620$. The translational and rotational motion, as well as concentration and dispersion, are compared with those of spheres of similar inertia. Disks and fibres both oversample high-speed fluid near the wall, in agreement with particle-resolved numerical simulations. Fibres tend to orient mostly in the streamwise direction while disks maintain their symmetry axis quasi-normal to the wall. This alignment is more stable for disks than for fibres: the latter undergo strong tumbling near the wall in response to the mean shear and turbulent fluid velocity fluctuations, whereas the former wobble about their preferential wall-normal orientation. The translational and rotational accelerations indicate that, despite the nominal relaxation times being similar, the disks are slower than the fibres in responding to wall turbulence. For both, wall contact causes strong and intermittent tumbling. The concentration profiles follow Rouse–Prandtl theory over a limited portion of the boundary layer, deviating near the wall and in the outer region. This is largely due to the non-uniform settling velocity, which decreases steeply approaching the wall for all particle types. This is, in turn, a consequence of the reduced particle diffusivity, which closely matches the profile of the eddy viscosity.

We investigate the statistics of rogue waves occurring in the inverse cascade of surface gravity wave turbulence. In such statistically homogeneous, stationary and isotropic wave fields, low-frequency waves are generated by nonlinear interactions rather than directly forced by a wave maker. This provides a laboratory realization of arguably the simplest nonlinear sea state, in which long-time acquisitions are performed and compared with theoretical models. The analysis of thousands of rogue waves reveals that some of their properties crucially depend on four-wave resonant interactions, large crests being for instance more likely than predicted by second-order models.

In this paper, the instability of shallow-water shear flow with a sheared parallel magnetic field is studied. Waves propagating in such magnetic shear flows encounter critical levels where the phase velocity relative to the basic flow, $c-U(y)$, matches the Alfvén wave velocities $\pm B(y)/\sqrt {\mu \rho }$, based on the local magnetic field $B(y)$, the magnetic permeability $\mu$, and the mass density of the fluid $\rho$. It is shown that when the two critical levels are close to each other, the critical layer can generate an instability. The instability problem is solved, combining asymptotic solutions at large wavenumbers and numerical solutions, and the mechanism of instability explained using the conservation of momentum. For the shallow-water magnetohydrodynamic system, the paper gives the general form of the local differential equation governing such coalescing critical layers for any generic field and flow profiles, and determines precisely how the magnetic field modifies the purely hydrodynamic stability criterion based on the potential vorticity gradient in the critical layer. The curvature of the magnetic field profile, or equivalently the electric current gradient $J' = - B''/\mu$ in the critical layer, is found to play a complementary role in the instability.

The atomization of a water column by a gas jet flow (Reynolds number $\sim O(10^{4}\unicode{x2013}10^{5})$) issued from a two-stage annular nozzle is investigated experimentally. Varying the nozzle geometry, the momentum flux ratio of the upper and lower jets, and the water flow rate, we measure the processes of atomization with high-speed imaging, analysed analytically into four regimes. In the bulk atomization regime, the atomization is driven by the lower jet, but it is forced to occur earlier by the stronger upper jet before the water column reaches the lower jet in the droplet atomization regime. Interestingly, the size of the atomized droplets remains unaffected by the momentum flux ratio of upper to lower jets. The atomization process is governed by the Rayleigh–Taylor instability, by which the estimated droplet size agrees well with the measurement. In the backflow regime, a strong reverse flow is induced to force a substantial portion of atomized droplets to be drawn backward to the nozzle; a floating liquid column regime is captured transitionally, i.e. the column stagnates near the lower nozzle when the water flow rate is very low. To understand the mechanisms of each regime, the single-phase jet flow is measured separately using particle image velocimetry, and implemented into the control volume analysis with which we predicted analytically and validated the conditions for the occurrence of each regime. It is found that the acceleration of gas flow (velocity gradient) experienced by the falling water is the key parameter to drive the atomization.

A scaling theory for the passive scalar transport in Couette flow, i.e. the flow between two parallel plates moving with different velocities, is proposed. This flow is determined by the bulk Reynolds number $Re_b$ and the Prandtl number $Pr$. In the turbulent regime, for moderate shear Reynolds number $Re_{\tau }$ and moderate $Pr$, we derive that the passive scalar transport characterised by the Nusselt number $Nu$ scales as $Nu \sim Pr^{1/2}Re_{\tau }^{2}Re_b^{-1}$. We then use the well-established scaling for the friction coefficient $C_f \sim Re_b^{-1/4}$ (corresponding to a shear Reynolds number $Re_{\tau } \sim Re_b^{7/8}$) which holds reasonably well within the range $3\times 10^{3} \leqslant Re_b \leqslant 10^{5}$, to obtain $Nu \sim Pr^{1/2}Re_b^{3/4}$ for the Nusselt number scaling. The theoretical results are tested against direct numerical simulations of Couette flows for the parameter ranges $81 \leqslant Re_b \leqslant 22361$ and $0.1 \leqslant Pr \leqslant 10$, finding good agreement. Analyses of the numerically obtained turbulent flow fields confirm logarithmic mean wall-parallel profiles of the velocity and the passive scalar in the inertial sublayer.

Debris flows are particle–fluid mixtures that pose a significant hazard to many communities throughout the world. Bouldery debris flows are often characterised by a deep dry granular flow front, which is followed by a progressively thinner and increasingly watery tail. The formation of highly destructive bouldery wave fronts is usually attributed to particle-size segregation. However, the moving-bed flume experiments of Davies (N. Z. J. Hydrol., vol. 29, 1990, pp. 18–46) show that discrete surges with dry fronts and watery tails also form in monodisperse particle–fluid mixtures. These observations motivate the development of a new depth-averaged mixture theory for debris flows, which explicitly takes account of the differing granular and phreatic surfaces, velocity shear, and relative motion between grains and fluid to explain these phenomena. The theory consists of four coupled conservation laws that describe the spatial and temporal evolution of the grain and water thicknesses and depth-averaged velocities. This system enables travelling wave solutions to be constructed that consist of (i) a large amplitude dry flow front that smoothly transitions to (ii) an undersaturated body, (iii) an oversaturated region and then (iv) a pure water tail. It is shown that these solutions are in good quantitative agreement with Davies’ experiments at high bed speeds and slope inclinations. At lower bed speeds and inclinations, the theory produces travelling wave solutions that connect to a steady-uniform upstream flow, and may or may not have a bulbous flow front, consistent with Davies’ observations.

We perform a three-dimensional short-wavelength linear stability analysis of numerically simulated two-dimensional Kelvin–Helmholtz vortices in homogeneous and stratified environments at a fixed Reynolds number of $Re = 300$. For the homogeneous case, the elliptic instability at the vortex core dominates at early times, before being taken over by the hyperbolic instability at the vortex edge. For the stratified case of Richardson number $ Ri = 0.08$, the early-time instabilities comprise a dominant elliptic instability at the core and a hyperbolic instability influenced strongly by stratification at the vortex edge. At intermediate times, the local approach shows a new branch of (convective) instability that emerges at the vortex core and subsequently moves towards the vortex edge. A few more convective instability bands appear at the vortex core and move away, before coalescing to form the most unstable region inside the vortex periphery at large times. In addition, the stagnation point instability is also recovered outside the periphery of the vortex at intermediate times. The dominant instability characteristics from the local approach are shown to be in good qualitative agreement with the results based on global instability studies for both homogeneous and stratified cases. A systematic study of the dependence of the dominant instability characteristics on $ Ri $ is then presented. While $ Ri = 0.1$ is identified as most unstable (with convective instability being dominant), another growth rate maximum at $ Ri = 0.025$ is not far behind (with the hyperbolic instability influenced by stratification being dominant). Finally, the local stability approach is shown to predict the potential orientation of the flow structures that would result from hyperbolic and convective instabilities, which is found to be consistent with three-dimensional numerical simulations reported previously.

Opposition control of artificially initiated turbulent spots in a laminar boundary layer was carried out in a low-turbulence wind tunnel with the aim to delay transition to turbulence by modifying the turbulent structure within the turbulent spots. The timing and duration of control, which was carried out using wall-normal jets from a spanwise slot, were pre-determined based on the baseline measurements of the transitional boundary layer. The results indicated that the high-speed region of the turbulent spots was cancelled by opposition control, which was replaced by a carpet of low-speed fluid. The application of the variable-interval time-averaging technique on the velocity fluctuation signals demonstrated a reduction in both the burst duration and intensity within the turbulent spots, but the burst frequency was increased.

The influence of thermocapillary stresses on motion of a slowly evaporating single-component droplet in Stokes flow is investigated analytically for the situation where the environment does not have a temperature gradient in the far field. The conservation equations are solved in the liquid and gas phases and coupled at the gas–liquid interface by applying conditions for conservation of mass, species, momentum and energy. It is found that thermocapillary stresses may influence droplet motion by changing the the interface velocity, and that the gas-phase Lewis number of the evaporating component determines whether Marangoni effects increase or decrease droplet drag. If the Lewis number is less than unity, then thermal Marangoni effects increase droplet drag, while if the Lewis number is greater than unity, then thermal Marangoni effects decrease droplet drag. This is related to the sign of the temperature gradient along the droplet surface that is induced by convection. It is found that conditions may exist where a vaporizing droplet in a microgravity environment will exhibit continuous translational motion driven by thermocapillary effects.

Earthquakes are a great challenge for the safety of nuclear reactors. To address this challenge, we need to better understand how the reactor core responds to seismic forcing. The reactor core is made of fuel assemblies, which are themselves composed of flexible fuel rods immersed in a strong axial flow. This gives rise to strongly coupled fluid–structure interactions whose accurate modelling generally requires high computational costs. In this paper, we introduce a new model able to capture the mechanical response of the reactor core subjected to seismic forcing with low computational costs. This model is based on potential flow theory for the fluid part, and Euler–Bernoulli beam theory for the structural part, allowing us to predict the response to seismic forcing in presence of axial flow. The linear equations are solved in the Fourier space to decrease computational time. For validation purposes, first we use the proposed model to compute the response of a single cylinder in axial flow. We then implement a multiple-cylinder geometry made of four fuel assemblies, each made of $8\times 8$ cylinders, corresponding to an experimental facility available at CEA. The comparison between numerical results and experiments shows good agreement. The model can predict correctly the added mass. It can also capture qualitatively the coupling between assemblies and the effect of confinement. This shows that a potential flow approach can give insight into the complex fluid–structure interactions within a nuclear reactor and, in particular, be used to predict the response to seismic forcing at low computational cost.

A theoretical model is developed for the forced convection film boiling phenomenon over a heated sphere moving vertically downwards in the water. Unprecedented compared with the previous analytical studies, this model accounts for the buoyancy effects while solving the momentum and energy equations in the vapour phase to obtain the velocity and the temperature distribution in terms of the vapour boundary layer thickness. To calculate the vapour boundary layer thickness, an energy balance is applied at the vapour–liquid interface. The flow of liquid around the sphere is considered to be governed by potential theory, and the energy equation in liquid is then solved for the known velocity distribution. We find that the vapour boundary layer thickness increases with an increase in the sphere temperature, the bulk water temperature and a decrease in the free stream velocity. This further results in a decrease in the film boiling heat transfer coefficient. The present study concludes that at low free stream velocities (${<}0.5\, {\rm m}\,{\rm s}^{-1}$) buoyancy becomes significant in delaying the separation, and when the velocity is further reduced the separation angle approaches $180^{\circ}$.

In this paper, two-phase flow simulations of oscillatory sheet flow experimental configurations involving medium and fine sand using a turbulence-resolving two-fluid model are presented. The turbulence-resolving two-phase flow model reproduces the differences of behaviour observed between medium and fine sand whereas turbulence-averaged models require an almost systematic tuning of empirical model coefficients for turbulence–particle interactions. The two-fluid model explicitly resolves these interactions and can be used to study in detail the differences observed experimentally. Detailed analysis of concentration profiles, flow hydrodynamics, turbulent statistics and vertical mass balance allowed the confirmation that unsteady effects, namely phase-lag effect and enhanced boundary layer thickness, for fine sand are not only due to the small settling velocity of the particles relative to the wave period. The occurrence and intensity of unsteady effects are also affected by a complex interplay between flow instabilities, strong solid-phase Reynolds stress and turbulence attenuation caused by the presence of the particles.

The transport coefficients of a dilute gas of inelastic hard spheres immersed in a gas of elastic hard spheres (molecular gas) are determined. We assume that the number density of the granular gas is much smaller than that of the surrounding molecular gas, so that the latter is not affected by the presence of the granular particles. In this situation, the molecular gas may be treated as a thermostat (or bath) of elastic hard spheres at a fixed temperature. The Boltzmann kinetic equation is the starting point of the present work. The first step is to characterise the reference state in the perturbation scheme, namely the homogeneous state. Theoretical results for the granular temperature and kurtosis obtained in the homogeneous steady state are compared against Monte Carlo simulations showing a good agreement. Then, the Chapman–Enskog method is employed to solve the Boltzmann equation to first order in spatial gradients. In dimensionless form, the Navier–Stokes–Fourier transport coefficients of the granular gas are given in terms of the mass ratio $m/m_g$ ($m$ and $m_g$ being the masses of a granular and a gas particle, respectively), the (reduced) bath temperature and the coefficient of restitution. Interestingly, previous results derived from a suspension model based on an effective fluid–solid interaction force are recovered in the Brownian limit ($m/m_g \to \infty$). Finally, as an application of the theory, a linear stability analysis of the homogeneous steady state is performed showing that this state is always linearly stable.

Solvent exchange is a process involving mixing between a good solvent with dissolved solute and a poor solvent. The process creates local oversaturation which causes the nucleation of minute solute droplets. Such ternary systems on a macro-scale have remained unexplored in the turbulent regime. We experimentally study the solvent exchange process by injecting mixtures of ethanol and trans-anethole into water, forming a turbulent buoyant jet in the upward direction. Locally, turbulent mixing causes oversaturation of the trans-anethole following turbulent entrainment. We optically measure the concentration of the nucleated droplets using a light attenuation technique and find that the radial concentration profile has a sub-Gaussian kurtosis. In contrast to the entrainment-based models, the spatial evolution of the oversaturation reveals continuous droplet nucleation downstream and radially across the jet, which we attribute to the limited mixing capacity of the jet. Although we are far from a full quantitative understanding, this work extends the knowledge on solvent exchange into the turbulent regime, and brings in a novel type of flow, broadening the scope of multicomponent, multiphase turbulent jets with phase transition.